LAB: Local linearity Name Topics and Skills: Functions, graphing Consider the graph of y =f(x) at the point (a, b) and imagine magnifying the boxed region of the graph around the point (a, b) (Figure 1a). Because the graph is smooth (meaning it has no sharp corners or cusps) near (a, b), the curve looks more like a straight line that also passes through (a, b) (Figure lb). As we zoom in further on the curve near (a, b), the line and curve become nearly indistinguishable. We say that the curve y =f(x) is locally linear at (a, 15) because the curve is approximately straight when examined microscopically (or locally) near (a, b). This special line passing through (a, b) that is approached by the curve as we zoom in is called the tangent line at (a, b). When we refer to the slope ofa curve at a given point (a, b), we mean the slope of the line tangent to the curve at (a, b). .1' (I (a) (b) Figure I l. The slope of a curve may be estimated by zooming in on the graph using a graphing utility until it appears linear. Suppose we wish to estimate the slope of the curve y = J; at (4, 2). a. Plot the graph of y = J; using the window [0, 8] x [0,4] (Figure 2a). Does the graph appear to be nearly linear in this graphing window? b. New zoom in by changing the graphing window to [3, 5] X [l 5,2.5] (Figure 2b). (If you are using a graphing calculator, the new window can be obtained automatically by using the zoom command, after rst setting the x and y zoom factors to 4.) Does the graph appear to be nearly linear near (4, 2) in this new graphing window? c. Two points on the curve, (3.5063291,1.8725195) and (4.4936709,2.119828), were obtained using the trace command on a graphing calculator (Figure 2b). The slope between these points is 2.119828 1.8725195 4.4936709 3.5063291 Find two other points on the curve, within the window [3,5] X [l .5, 2.5] and calculate the slope of the line = 02504791147. (1) joining these points. Approximate your coordinates with eight digits of accuracy. Is your answer close to the slope given in (1)? Explain. d. Zoom in again by graphing the function using the window [375,425] X [1 8712.125] Locate two points on the curve in this new window and compute the slope of the curve again with eight digits of accuracy. e. Make a reasonable estimate of the slope of the curve at (4, 2) based upon your work in parts (c) and (d)